psubclsubN - Mathbox for Norm Megill

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Theorem psubclsubN 40447

Description: A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
psubclsub.s	⊢ 𝑆 = (PSubSp‘𝐾)
psubclsub.c	⊢ 𝐶 = (PSubCl‘𝐾)

**Assertion**
Ref	Expression
psubclsubN	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑆)

**Proof of Theorem psubclsubN**
Step	Hyp	Ref	Expression
1		eqid 2741	. . 3 ⊢ (⊥_𝑃‘𝐾) = (⊥_𝑃‘𝐾)
2		psubclsub.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
3	1, 2	psubcli2N 40446	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) = 𝑋)
4		eqid 2741	. . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	4, 1, 2	psubcliN 40445	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) = 𝑋))
6	5	simpld 496	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ (Atoms‘𝐾))
7		psubclsub.s	. . . . . 6 ⊢ 𝑆 = (PSubSp‘𝐾)
8	4, 7, 1	polsubN 40414	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥_𝑃‘𝐾)‘𝑋) ∈ 𝑆)
9	6, 8	syldan 598	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘𝑋) ∈ 𝑆)
10	4, 7	psubssat 40261	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ((⊥_𝑃‘𝐾)‘𝑋) ∈ 𝑆) → ((⊥_𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
11	9, 10	syldan 598	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
12	4, 7, 1	polsubN 40414	. . 3 ⊢ ((𝐾 ∈ HL ∧ ((⊥_𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) ∈ 𝑆)
13	11, 12	syldan 598	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) ∈ 𝑆)
14	3, 13	eqeltrrd 2842	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ⊆ wss 3885 ‘cfv 6489 Atomscatm 39770 HLchlt 39857 PSubSpcpsubsp 40003 ⊥_𝑃cpolN 40409 PSubClcpscN 40441

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-iun 4926 df-iin 4927 df-br 5076 df-opab 5138 df-mpt 5157 df-id 5516 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-proset 18255 df-poset 18274 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p1 18385 df-lat 18393 df-clat 18460 df-oposet 39683 df-ol 39685 df-oml 39686 df-ats 39774 df-atl 39805 df-cvlat 39829 df-hlat 39858 df-psubsp 40010 df-pmap 40011 df-polarityN 40410 df-psubclN 40442

This theorem is referenced by: pclfinclN 40457

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